This is very interesting! See, I don't know very much about many languages, but in my previous (begun-but-never-finished) conlang, Cakuacakuán, I decided that I did not want to have a verb for "to be," and so a sentence such as "This is a bicycle," translated to "Ce abicikleta" (/tse əˌbitsiˈkɺetä/). That literally means "This a-bicycle." This was inspired by Russian. But then when I was constructing this language, which is based heavily off of Cakuacakuán, I realized that in the case that you are equating something to another noun or adjective, it loses the possibility of tense, so I had to think of a way to incorporate that and came up with the tense endings. I already had the adjective ending "-im" from Cakuacakuán.
A common way that languages deal with this is suppletion. For instance, future tense will be expressed with forms of a verb meaning "to become". There's also no reason why you couldn't do what languages which don't have tense as an inflection category do and simply use time adverbs. (E.g. "He at-one-time my father" = "He used to be my father".)
I think that's probably what would have happened had I continued with Cakuacakuán. The present tense wouldn't have a time adverb, but all other tenses would. As it is now, I will probably have some with this language as there are tenses I haven't thought about with my endings. As they are they are just basic indicators. I may have other auxiliary tense indicators to determine if something is continuing or not, but we'll see.